Baxter’s inequality for triangular arrays

Abstract

A central problem in time series analysis is prediction of a future observation. The theory of optimal linear prediction has been well understood since the seminal work of A. Kolmogorov and N. Wiener during World War II. A simplifying assumption is to assume that one-step-ahead prediction is carried out based on observing the infinite past of the time series. In practice, however, only a finite stretch of the recent past is observed. In this context, Baxter’s inequality is a fundamental tool for understanding how the coefficients in the finite-past predictor relate to those based on the infinite past. We prove a generalization of Baxter’s inequality for triangular arrays of stationary random variables under the condition that the spectral density functions associated with the different rows converge. Themotivating examples are statistical time series settings where the autoregressive coefficients are re-estimated as new data are acquired, producing new fitted processes — and new predictors — for each n.